I am now a postdoc at RWTH Aachen.

Here are the lecture notes from the course Math 110 (Linear algebra) that I taught in summer 2016. They are intended to accompany the third edition of Axler's Linear Algebra Done Right. Some students found these notes useful.

Thesis:

Computing modular forms for the Weil representation. pdf Includes some minor corrections.

Preprints:

29. On the non-existence of singular Borcherds products (with H. Wang). pdf

28. Binary theta series and Borcherds products (with M. Schwagenscheidt). pdf J. Number Theory 249 (2023) 441--461 link

27. Mathieu moonshine and Borcherds products (with H. Wang). pdf

26. The fake monster algebra and singular Borcherds products (with H. Wang). pdf

25. On Hermitian Eisenstein series of degree 2 (with A. Hauffe-Waschbüsch and A. Krieg), pdf Funct. Approx. Comment. Math. 68(1) (2023) 127--141 link

24. Modular forms with poles on hyperplane arrangements (with H. Wang). pdf

23. Siegel modular forms of degree two and level five (with H. Wang). pdf Ramanujan J. 60, 597--613 (2023) link

22. Graded rings of Hermitian modular forms with singularities (with H. Wang). pdf Manuscripta Math. 170, 283--311 (2023) link

21. Free algebras of modular forms on ball quotients (with H. Wang). pdf data

20. On weak Jacobi forms of rank two (with H. Wang). pdf To appear in J. Algebra.

19. Simple lattices and free algebras of modular forms (with H. Wang). pdf Adv. Math. 413 (2023), 108835 link

18. Projective spaces as orthogonal modular varieties (with H. Wang). pdf data To appear in Transform. Groups link

17. Borcherds products of half-integral weight (with H. Wang). pdf J. Number Theory 238 (2022) 944--950 link

16. On some free algebras of orthogonal modular forms (with H. Wang). pdf Adv. Math. 373 (2020), 107332 link

15. Two graded rings of Hermitian modular forms. pdf Relations for Q(sqrt(-11)) here. Abh. Math. Sem. Univ. Hamburg 91 (2021) 257--285 link

14. Higher pullbacks of modular forms on orthogonal groups. pdf Forum Math. 33 (2021) 631-652 link

13. Graded rings of paramodular forms of levels 5 and 7. pdf Relations for level 5 and level 7. Fourier expansions of generators here. J. Number Theory 209 (2020) 483-515 link

12. Twisted component sums of vector-valued modular forms (with M. Schwagenscheidt). pdf Abh. Math. Sem. Univ. Hamburg 89 (2019) 151-168 link

11. A construction of antisymmetric modular forms for Weil representations. pdf Math. Zeitschrift 296 (2020) 391--408 link

10. The rings of Hilbert modular forms for Q(sqrt29) and Q(sqrt37). pdf Data for Q(sqrt29) and Q(sqrt37) With an appendix by A. Logan: pdf magma files. J. Algebra 559 (2020), 679-711. link

9. Remarks on the theta decomposition of vector-valued Jacobi forms. pdf J. Number Theory 197 (2019), 250-267. link

8. Short proof of Rademacher's formula for partitions (with W. Pribitkin). pdf Res. Number Theory 5 (2019), no. 2, Art. 17, 6 pp. link

7. Rankin-Cohen brackets and Serre derivatives as Poincaré series. pdf Res. Number Theory 4 (2018), no. 4, Art. 37, 13 pp. link

6. A p-adic completion of Zagier's Eisenstein series. pdf

5. Vector-valued Hirzebruch-Zagier series and class number sums. pdf Res. Math. Sci. 5 (2018), no. 2, Paper No. 25, 13 pp. link

4. Overpartition M2-rank differences, class number relations, and vector-valued mock Eisenstein series. pdf Acta Arith. 189 (2019), no. 4, 347-365. link

3. Poincaré square series of small weight. pdf Ramanujan J. 48 (2019), no. 3, 585-612. link

2. Vector-valued Eisenstein series of small weight. pdf Int. J. Number Theory 15 (2019), no. 2, 265-287. link

1. Poincaré square series for the Weil representation. pdf Ramanujan J. 47 (2018), no. 3, 605-650. link

Programs:

weilrep is a Sage file for calculations with vector-valued modular forms for Weil representations, Jacobi forms, theta lifts and Borcherds products. Download it from GitHub. Please see the readme for an explanation of how to use it.

If you find a bug in this program or have other questions or comments please email me at brandon.williams (at) mathA (dot) rwth-aachen (dot) de

Here is some code for working with eta products. It computes the q-series of an eta product and tries to identify an eta product from a q-series. Some examples are at the end of the worksheet.

Here is some code for computing exact values of Dedekind zeta functions for totally real number fields.